Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
GEOMETRY
Lesson 7: Equations for Lines Using Normal Segments
Student Outcomes

Students recognize a segment perpendicular to a line, with one of its endpoints on the line as a normal
segment.

Students recognize that when a line and a normal segment intersect at the origin, the segment from (0,0) to
(𝑎1 , 𝑎2 ) is the normal segment, with a slope of
of –
𝑎1
.
𝑎2
𝑎2
𝑎1
, and the equation of the line is 𝑎1 𝑥 + 𝑎2 𝑦 = 𝑐 with a slope
Lesson Notes
This lesson focuses on MP.4 because students work extensively to model robot behavior using coordinates.
Classwork
Opening Exercise (5 minutes)
This exercise can be modeled by the teacher with the whole class, given to groups to present solutions to the class, or
used as a supplement to the lesson.
Opening Exercise
The equations given are in standard form. Put each equation in slope-intercept form. State the slope and the
𝒚-intercept.
1.
𝟔𝒙 + 𝟑𝒚 = 𝟏𝟐
2.
𝟓𝒙 + 𝟕𝒚 = 𝟏𝟒
𝒚 = −𝟐𝒙 + 𝟒
𝟓
𝒚= − 𝒙+𝟐
𝟕
slope = −𝟐
slope = −
𝒚-intercept = 𝟒
𝒚-intercept = 𝟐
𝟓
𝟕
3.
𝟐𝒙 − 𝟓𝒚 = −𝟕
𝒚=
𝟐
𝟕
𝒙+
𝟓
𝟓
slope =
𝟐
𝟓
𝒚-intercept =
Scaffolding:
𝟕
𝟓
Provide visuals to reinforce
standard and slope-intercept
forms:
 Standard form is
𝐴𝑥 + 𝐵𝑦 = 𝐶 where 𝐴, 𝐵,
and 𝐶 are integers.
 Slope-intercept form is
𝑦 = 𝑚𝑥 + 𝑏 where 𝑚 is
the slope, and 𝑏 is the 𝑦intercept.
Lesson 7:
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
72
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
GEOMETRY
Discussion (20 minutes)
Let’s revisit the robot from Lesson 4. Recall that it is moving along the line
𝑦 = 3𝑥 − 600. At the point (400, 600), it detected the loudest “ping,” and the
programmers had the robot change direction at that point and move along a linear path
that was perpendicular to its original path.
Scaffolding:
Plotting points helps students
visualize the problems and
understand the translations
required.
At the end of Lesson 6, students theorized that the beacon might lie on the 𝑥-axis. If it did, it would be located at the
point (2200, 0).

How did we determine the location of the beacon?
We translated point 𝐹 to the origin and substituted the coordinates of the translated points 𝐸′and 𝐺′
into the formula for the condition of perpendicularity 𝑎1 𝑏1 + 𝑎2 𝑏2 = 0, using 𝑔 − 400 as the
𝑥-coordinate of 𝐺 ′ . We solved the equation to get 𝑔 = 2200.
Let’s push this idea a little further. Suppose 𝑃(𝑥, 𝑦) is any point on the line containing ̅̅̅̅
𝐸𝐹 . What can you say
̅̅̅̅ and ̅̅̅̅
about 𝐹𝑃
𝐹𝐺 ?




Using 𝐹(400, 600), 𝐺(2200, 0), and 𝑃(𝑥, 𝑦), translate the points so that 𝐹 is at the origin. What are the new
coordinates?


They are also perpendicular.
If we translate all points in the figure using the translation vector ⟨−400, −600⟩ that takes 𝐹 to the
origin, then 𝐹′ is (0, 0), 𝐺 becomes 𝐺′(1800, −600), and 𝑃 becomes 𝑃′(𝑥 − 400, 𝑦 − 600).
If the condition for perpendicularity is 𝑎𝑏 + 𝑐𝑑 = 0, how could we write an equation involving 𝑥 and 𝑦?
 The condition that ̅̅̅̅̅
𝐹′𝐺′ and ̅̅̅̅̅
𝐹′𝑃′ are perpendicular becomes:
1800(𝑥 − 400) + (−600)(𝑦 − 600) = 0
1800𝑥 − 720,000 − 600𝑦 + 360,000 = 0
1800𝑥 − 600𝑦 = 360,000.

This is the equation of the line in standard form: 𝐴𝑥 + 𝐵𝑦 = 𝐶. Look at the values of 𝐴 and 𝐵 and at the work
above. Do you see a relationship between 𝐴 and 𝐵 and the work above?

The coordinates of 𝐺′ (the translation of 𝐺) are (1800, −600), which are the same values as 𝐴 and 𝐵.
Lesson 7:
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
73
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
GEOMETRY


Let’s say that in a more specific way. I will state the relationship, and you repeat it and explain it to your
partner.

𝐴 is the 𝑥-coordinate (abscissa) of the image of the point of the perpendicular segment that does not lie
on the line when the point of perpendicularity is at the origin.

𝐵 is the 𝑦-coordinate (ordinate) of the image of the point of the perpendicular segment that does not
lie on the line when the point of perpendicularity is at the origin.
Now, let’s put the equation in slope-intercept form.

−600𝑦 = −1800𝑥 + 360,000
𝑦 = 3𝑥 − 600, which is the equation of ⃡𝐸𝐹 .

MP.7
What does the equation you wrote represent?
 ̅̅̅̅
𝐹𝑃 will be perpendicular to ̅̅̅̅
𝐹𝐺 as long as point 𝑃 lies on the line containing ̅̅̅̅
𝐸𝐹 , which is given by the
equation 𝑦 = 3𝑥 − 600.
In the next part of this lesson, students generalize what they just discovered. This can be done in several ways.
1.
Present the question, and explain that they are trying to make this process work for any points with coordinates
𝐴(𝑎, 𝑏), 𝐵(𝑐, 𝑑), and 𝑃(𝑥, 𝑦). Allow time for students to think and talk to their neighbors for a few minutes; then,
show the diagrams and give them more time to talk. Finally, pull everyone together, and discuss each step as a
class.
2.
Assign some groups the task with no leading questions, and let them work independently while other groups are
getting different levels of help, some even being directly instructed by the teacher.

How can we generalize this finding?

Given point 𝐴(𝑎, 𝑏), which lies on line 𝑙; point 𝐵(𝑐, 𝑑) not on line 𝑙; and ̅̅̅̅
𝐴𝐵 perpendicular to line 𝑙, then any
̅̅̅̅ ⊥ ̅̅̅̅
point 𝑃(𝑥, 𝑦) on line 𝑙 will satisfy the relationship 𝐴𝑃
𝐴𝐵 . Draw the picture described.
𝑩
𝑙
(𝒄, 𝒅)
(𝒙, 𝒚)
𝑷
𝑨
Lesson 7:
(𝒂, 𝒃)
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
74
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
GEOMETRY

Translate the points. Which point should be on the origin? What is the translation used? What are the
coordinates of the translated points?

𝐴 is the common point and should be translated to the origin. The translation is ⟨−𝑎, −𝑏⟩, or left
𝑎 𝑢𝑛𝑖𝑡𝑠 and down 𝑏 𝑢𝑛𝑖𝑡𝑠. The translated points are 𝐴′(0, 0), 𝐵′(𝑐 − 𝑎, 𝑑 − 𝑏), and 𝑃′(𝑥 − 𝑎, 𝑦 − 𝑏).
𝑙
𝑩′
(𝒄 − 𝒂, 𝒅 − 𝒃)
𝑷′
(𝒙 − 𝒂, 𝒚 − 𝒃)
𝑨′

If the segments are perpendicular, write the equation that must hold true.




(𝟎, 𝟎)
(𝑐 − 𝑎)(𝑥 − 𝑎) + (𝑑 − 𝑏)(𝑦 − 𝑏) = 0
If 𝐴 = (𝑐 − 𝑎) and 𝐵 = (𝑑 − 𝑏), write this equation substituting in 𝐴 and 𝐵. Which line have we written the
equation of?
Scaffolding:
 We end up with the equation of the line that passes through
 Have students leave the equations
point 𝐴 that is perpendicular to ̅̅̅̅
𝐴𝐵 : 𝐴(𝑥 − 𝑎) + 𝐵(𝑦 − 𝑏) = 0.
in standard form.
What do 𝐴 and 𝐵 represent graphically?
 Provide these steps:
 𝐴 is the abscissa, and 𝐵 is the ordinate of the image of point 𝐵.
(8 − 5)(𝑥 − 5) + (2 + 7)(𝑦 + 7) = 0
We call ̅̅̅̅
𝐴𝐵 a normal segment to line 𝑙 because it has one endpoint on
the line and is perpendicular to the line.
Simplify the parentheses without
variables.
 Explain to your neighbor what a normal segment is, and write
your own definition.
DEFINITION: A line segment with one endpoint on a line and perpendicular to the
line is called a normal segment to the line.
3(𝑥 − 5) + 9(𝑦 + 7) = 0
Separate the variables by putting 𝑥
on one side and 𝑦 on the other.
9(𝑦 + 7) = −3(𝑥 − 5)
Distribute the coefficients.
9𝑦 + 63 = −3𝑥 + 15
Bring the constant on the right to
the left.
9𝑦 = −3𝑥 − 48
Divide by the coefficient of 𝑦.
3
48
𝑦=− 𝑥−
9
9
Simplify if necessary.
1
16
𝑦=− 𝑥−
3
3
Lesson 7:
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
75
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7
M4
GEOMETRY
Example (5 minutes)
Example
Given 𝑨(𝟓, −𝟕) and 𝑩(𝟖, 𝟐):
a.
̅̅̅̅.
Find an equation for the line through 𝑨 and perpendicular to 𝑨𝑩
𝑨′ (𝟎, 𝟎), 𝑩′ (𝟖 − 𝟓, 𝟐 − (−𝟕)), and 𝑷′ (𝒙 − 𝟓, 𝒚 − (−𝟕))
(𝟖 − 𝟓)(𝒙 − 𝟓) + (𝟐 + 𝟕)(𝒚 + 𝟕) = 𝟎
𝟑(𝒙 − 𝟓) = −𝟗(𝒚 + 𝟕)
𝟑𝒙 − 𝟏𝟓 = −𝟗𝒚 − 𝟔𝟑
𝟏
𝟏𝟔
𝒚= − 𝒙−
𝟑
𝟑
b.
̅̅̅̅.
Find an equation for the line through 𝑩 and perpendicular to 𝑨𝑩
𝑩′ (𝟎, 𝟎), 𝑨′ (𝟓 − 𝟖, −𝟕 − 𝟐), and 𝑷′(𝒙 − 𝟖, 𝒚 − 𝟐)
(𝟓 − 𝟖)(𝒙 − 𝟖) + (−𝟕 − 𝟐)(𝒚 − 𝟐) = 𝟎
−𝟑(𝒙 − 𝟖) = 𝟗(𝒚 − 𝟐)
−𝟑𝒙 + 𝟐𝟒 = 𝟗𝒚 − 𝟏𝟖
𝟏
𝟏𝟒
𝒚=− 𝒙+
𝟑
𝟑
Exercises (8 minutes)
Have students plot points to aid in solving problems.
Exercises
1.
Given 𝑼(−𝟒, −𝟏) and 𝑽(𝟕, 𝟏):
a.
̅̅̅̅.
Write an equation for the line through 𝑼 and perpendicular to 𝑼𝑽
𝑼′ (𝟎, 𝟎), 𝑽′ (𝟕 − (−𝟒), 𝟏 − (−𝟏)), 𝑷′ (𝒙 − (−𝟒), 𝒚 − (−𝟏))
(𝟕 + 𝟒)(𝒙 + 𝟒) + (𝟏 + 𝟏)(𝒚 + 𝟏) = 𝟎
𝟏𝟏𝒙 + 𝟒𝟒 = −𝟐𝒚 − 𝟐
𝟏𝟏
𝒚=−
𝒙 − 𝟐𝟑
𝟐
b.
̅̅̅̅.
Write an equation for the line through 𝑽 and perpendicular to 𝑼𝑽
𝑽′ (𝟎, 𝟎), 𝑼′ (−𝟒 − 𝟕, −𝟏 − 𝟏), 𝑷′(𝒙 − 𝟕, 𝒚 − 𝟏)
(−𝟕 − 𝟒)(𝒙 − 𝟕) + (−𝟏 − 𝟏)(𝒚 − 𝟏) = 𝟎
−𝟏𝟏𝒙 + 𝟕𝟕 = 𝟐𝒚 − 𝟐
𝟏𝟏
𝟕𝟗
𝒚=−
𝒙+
𝟐
𝟐
Lesson 7:
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
76
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7
M4
GEOMETRY
2.
Given 𝑺(𝟓, −𝟒) and 𝑻(−𝟖, 𝟏𝟐):
a.
Write an equation for the line through 𝑺 and perpendicular to ̅̅̅̅
𝑺𝑻.
𝑺′ (𝟎, 𝟎), 𝑻′ (−𝟖 − 𝟓, 𝟏𝟐 − (−𝟒)), 𝑷′ (𝒙 − 𝟓, 𝒚 − (−𝟒))
(−𝟖 − 𝟓)(𝒙 − 𝟓) + (𝟏𝟐 − (−𝟒))(𝒚 − (−𝟒)) = 𝟎
−𝟏𝟑𝒙 + 𝟔𝟓 = −𝟏𝟔𝒚 − 𝟔𝟒
𝟏𝟑
𝟏𝟐𝟗
𝒚=
𝒙−
𝟏𝟔
𝟏𝟔
b.
Write an equation for the line through 𝑻 and perpendicular to ̅̅̅̅
𝑺𝑻.
𝑻′ (𝟎, 𝟎), 𝑺′ (𝟓 − (−𝟖), −𝟒 − 𝟏𝟐), 𝑷′(𝒙 − (−𝟖), 𝒚 − 𝟏𝟐)
(𝟓 − (−𝟖)) (𝒙 − (−𝟖)) + (−𝟒 − 𝟏𝟐)(𝒚 − 𝟏𝟐) = 𝟎
𝟏𝟑𝒙 + 𝟏𝟎𝟒 = 𝟏𝟔𝒚 − 𝟏𝟗𝟐
𝟏𝟑
𝟑𝟕
𝒚=
𝒙+
𝟏𝟔
𝟐
Closing (2 minutes)
Describe the characteristics of a normal segment.
A line segment with one endpoint on a line and perpendicular to the line is called a normal segment to the line.
Every equation of a line through a given point (𝒂, 𝒃) has the form 𝑨(𝒙 − 𝒂) + 𝑩(𝒚 − 𝒃) = 𝟎. Explain how the values of
𝑨 and 𝑩 are obtained.
𝑨 is the value of the abscissa of the image of the endpoint of the normal segment that does not lie on the line after the
figure has been translated so that the image of the endpoint that does lie on the line, the point of perpendicularity, is at
the origin.
𝑩 is the value of the ordinate of the image of the endpoint of the normal segment that does not lie on the line after the
figure has been translated so that the image of the endpoint that does lie on the line, the point of perpendicularity, is at
the origin.
Exit Ticket (5 minutes)
Lesson 7:
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
77
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
GEOMETRY
Name
Date
Lesson 7: Equations for Lines Using Normal Segments
Exit Ticket
Given 𝐴(−5, −3), 𝐵(−1, 6), and 𝐶(𝑥, 𝑦):
a.
What are the coordinates of the translated points if 𝐵 moves to the origin?
b.
Write the condition for perpendicularity equation.
c.
Write the equation for the normal line in slope-intercept form.
Lesson 7:
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
78
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7
M4
GEOMETRY
Exit Ticket Sample Solutions
Given 𝑨(−𝟓, −𝟑), 𝑩(−𝟏, 𝟔), and 𝑪(𝒙, 𝒚):
a.
What are the coordinates of the translated points if 𝑩 moves to the origin?
𝑨′(−𝟒, −𝟗), 𝑩′(𝟎, 𝟎), 𝑪′(𝒙 + 𝟏, 𝒚 − 𝟔)
b.
Write the condition for perpendicularity equation.
−𝟒(𝒙 + 𝟏) − 𝟗(𝒚 − 𝟔) = 𝟎
c.
Write the equation for the normal line in slope-intercept form.
𝟒
𝟓𝟎
𝒚=− 𝒙+
𝟗
𝟗
Problem Set Sample Solutions
1.
Given points 𝑪(−𝟒, 𝟑) and 𝑫(𝟑, 𝟑):
a.
̅̅̅̅.
Write the equation of the line through 𝑪 and perpendicular to 𝑪𝑫
𝒙 = −𝟒
b.
Write the equation of the line through 𝑫 and perpendicular to ̅̅̅̅
𝑪𝑫.
𝒙=𝟑
2.
Given points 𝑵(𝟕, 𝟔) and 𝑴(𝟕, −𝟐):
a.
̅̅̅̅̅.
Write the equation of the line through 𝑴 and perpendicular to 𝑴𝑵
𝒚 = −𝟐
b.
Write the equation of the line through 𝑵 and perpendicular to ̅̅̅̅̅
𝑴𝑵.
𝒚=𝟔
3.
The equation of a line is given by the equation 𝟖(𝒙 − 𝟒) + 𝟑(𝒚 + 𝟐) = 𝟎.
a.
What are the coordinates of the image of the endpoint of the normal segment that does not lie on the line?
Explain your answer.
(𝟖, 𝟑) because 𝑨(𝒙 – 𝟒) + 𝑩(𝒚 + 𝟐) = 𝟎 is the original formula, and (𝑨, 𝑩) are the coordinates of the image
of the endpoint of the normal segment not on the line.
b.
What translation occurred to move the point of perpendicularity to the origin?
⟨−𝟒, 𝟐⟩, or left 𝟒 units up 𝟐 units
Lesson 7:
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
79
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
GEOMETRY
c.
What were the coordinates of the original point of perpendicularity? Explain your answer.
(𝟒, −𝟐) because the translation of ⟨−𝟒, 𝟐⟩ was required to move the point of perpendicularity to the origin.
d.
What were the endpoints of the original normal segment?
𝟖= 𝒄−𝟒
and
𝒄 = 𝟏𝟐
𝟑 = 𝒅 − (−𝟐)
𝒅=𝟏
The endpoints of a normal segment to the given line are 𝑨(𝟒, −𝟐) and 𝑩(𝟏𝟐, 𝟏).
4.
A coach is laying out lanes for a race. The lanes are perpendicular to a segment of the track such that one endpoint
of the segment is (𝟐, 𝟓𝟎), and the other is (𝟐𝟎, 𝟔𝟓). What are the equations of the lines through the endpoints?
𝟔
𝟕𝟖𝟔
𝟔
𝒚= − 𝒙+
; 𝒚 = − 𝒙 + 𝟖𝟗
𝟓
𝟏𝟓
𝟓
Lesson 7:
Equations for Lines Using Normal Segments
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M4-TE-1.3.0-09.2015
80
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.